[1337isnot1337]

Change and mathematical scope

Removing the productive controller's large idle bill needs a changed architecture. This package proves two global obstructions for binary controllers with independent native stage multipliers, arbitrary constant changes of basis, and independent monomial Fourier outputs. No new supplier or κ is claimed.

• An all-full controller on p main pairs needs at least 3p native calls. Its target-minus-paid moment per full call cannot exceed disjoint three-call units. The proof uses simultaneous native-projector diagonalization and an edge-parity count.
• Every such controller needs at least 2p+1 query-bearing calls. At exactly that count and p≥2, at most two calls can be full. For the retained BIT profile, the complete target-minus-paid moment is at most −15 for every exponent in [0,1]. The proof uses the corank-one simplex forced by a shortest rank-one nilpotent factorization, then commuting-projector constraints. The uniform moment rejection reduces to an exact integer inequality.

Start with GLOBAL-PROOF.md. The result leaves mixed controllers with additional queries, changed central queries/output contracts, dependent stage multipliers and new local families open. It does not reduce arbitrary complex dyadic operators modulo two or establish a universal native lower bound.

The complete four-field/five-call discovery census is also retained: 1,680 native maps and 105 thin-query maps after arbitrary binary changes of basis; 12,480 query-cancellation survivors, all rejected by exact independent-output factorization. The global proofs extend beyond that finite census.

Task and overlap

Explicit user assignment: seek a much higher K through a bottleneck breakthrough and publish verified checkpoints. Scope changes from the historical complex-new-helper-inventory card to controller architecture. Scientific base and final refreshed canonical main: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Priority revision 7; selected conditional κ remains 0.00302015.

The source-bound #4440/#4470 interface is retained unchanged. #4602 identifies the idle/query budget; #4603 excludes particular fixed-window rewrites; #4623 changes to a border endpoint and prices recovery. This package instead permits arbitrary changes of basis and proves the two no-idle exclusions for every pair count under the stated exact-output contract. A refreshed live search covered 436 open PR bodies plus the current handoff thread and recent discussion index. It is an overlap screen, not a literature-wide novelty certificate.

Validation

python3 -B scripts/research_package.py --package research/productive-controller-basis-search
python3 -B scripts/check_research_commit.py --package research/productive-controller-basis-search --check-source-history

Registered replay passes locally on Linux/Python 3.12 with a C++17 compiler. The 13-file committed export passes integrity checks and a fresh Git-free replay under Python -O. GitHub plan and package replay both passed at exact head 08ac39202263203fded429c826ba9228ac9df803. It recompiles and reruns all 16 finite masks, independently replays every survivor as a polynomial, checks 128 optimized/numeric transitions against separate symbolic evaluation, verifies all 1,680 native gate relations, and compares the exact receipt. The global proof controls cover 47,904 simplex pairings, 528 forced projectors and 992 endpoint obstructions through 32 pairs, the complete two-coordinate nilpotent parity table, the n=2 exception, a working six-call word and an actual sharp two-native-call example. The all-t BIT bound uses exact integer arithmetic.

Written proofs establish the all-pair-count statements; those finite controls support the implementation and do not constitute formal-kernel or independent human verification. No selected cold supplier replay or outer theorem replay is claimed because no new supplier is proposed. Frozen source archives, selected certificates and production generators are unchanged.

Credit

Credit Cinna Davis's #4440/#4470 native/query and helper work, zxv's productive supplier/compiler lineage, Jacob Alan Sussman's mixed-target recursion, and the retained BIT, Clifford, precision and outer contributors. The rank-factorization and elementary nilpotent identities are used as mathematical tools; no literature-wide novelty claim is made. New proofs, implementation and writing by Cinna Davis / 1337isnot1337 with substantial OpenAI Codex assistance. No subagents. Apache-2.0.

Open next questions

Construct a mixed controller with more than 2p+1 queries, or change the central query/output contract, then test every coefficient and price the complete invoice. Repeating a shortest-query or all-full no-idle search under the same hypotheses is excluded. No such profitable changed controller is supplied here; a larger K remains the campaign objective.